Fast SDP Relaxation of the Optimal Power Flow Using the Line-Wise Model for Representing Meshed Transmission Networks

Abstract

In this paper, we propose a novel Semidefinite Programming (SDP) relaxation of the Optimal Power Flow (OPF) problem. The proposed formulation utilizes the Line-Wise Model (LWM) to represent meshed transmission networks. This allows the constraints of the proposed formulation to mostly depend on diagonal elements of the voltages&#x0027; Hermitian <inline-formula><tex-math notation="LaTeX">${\bm{W}}$</tex-math></inline-formula> matrix in contrast to the Bus Injection Model (BIM) where constraints heavily utilize its off-diagonal elements. Test cases with bus sizes of 3 to 9241 were considered and chordal sparsity was exploited for them. Obtained results show that the proposed SDP-LW OPF formulation manages to provide solutions of similar or better quality for most test cases that belong to the Typical and Congested (TYP and API) Operating conditions. Furthermore, the proposed SDP-LW OPF formulation manages to reduce the solver&#x0027;s computational time for most of large-scale test cases with reductions up to 80.298&#x0025;. Results analysis shows that reductions in the solver&#x0027;s computational time upon using the proposed SDP-LW OPF formulation are affected by its ability to significantly reduce constraints with off-diagonal elements of the <inline-formula><tex-math notation="LaTeX">${\bm{W}}$</tex-math></inline-formula> matrix without drastically increasing constraints with diagonal elements of the <inline-formula><tex-math notation="LaTeX">${\bm{W}}$</tex-math></inline-formula> matrix. Furthermore, the percentage of large sized cliques is found to affect the obtained reductions through its influence over the number of needed linking constraints for relating the elements of the <inline-formula><tex-math notation="LaTeX">${\bm{W}}$</tex-math></inline-formula> matrices of decomposed cliques.